## Banach Journal of Mathematical Analysis

### Linear maps preserving pseudospectrum and condition spectrum

#### Abstract

We discuss properties of pseudospectrum and condition spectrum of an element in a complex unital Banach algebra and its $\epsilon$-perturbation. Several results are proved about linear maps preserving pseudospectrum/ condition spectrum. These include the following: (1) Let $A, B$ be complex unital Banach algebras and $\epsilon$ is positive. Let $\Phi: A\rightarrow B$ be an $\epsilon$-pseudospectrum preserving linear onto map. Then $\Phi$ preserves spectrum. If $A$ and $B$ are uniform algebras, then, $\Phi$ is an isometric isomorphism. (2) Let $A, B$ be uniform algebras and $\epsilon \in (0,1)$. Let $\Phi:A\rightarrow B$ be an $\epsilon$-condition spectrum preserving linear map. Then $\Phi$ is an $\epsilon^{'}$-almost multiplicative map, where $\epsilon, \epsilon^{'}$ tend to zero simultaneously.

#### Article information

Source
Banach J. Math. Anal., Volume 6, Number 1 (2012), 45-60.

Dates
First available in Project Euclid: 14 May 2012

https://projecteuclid.org/euclid.bjma/1337014664

Digital Object Identifier
doi:10.15352/bjma/1337014664

Mathematical Reviews number (MathSciNet)
MR2862542

Zentralblatt MATH identifier
1258.47055

#### Citation

Krishna Kumar, G.; Kulkarni, S. H. Linear maps preserving pseudospectrum and condition spectrum. Banach J. Math. Anal. 6 (2012), no. 1, 45--60. doi:10.15352/bjma/1337014664. https://projecteuclid.org/euclid.bjma/1337014664

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